Abstract
In the preceding chapters, we witnessed the new aspects of physics brought about by Einstein’s Relativity. All of this arose because: a) there is a speed limit in nature, \(c\), the maximum speed at which influences can be propagated and b) because all inertial observers are physically equivalent, they must all agree on this speed limit. However, not a word was mentioned about gravity and it is with the inclusion of gravity that Einstein’s Relativity takes on a whole new and exciting complexion. Einstein’s Relativity without gravity is called “Special Relativity” to distinguish it from Relativity with gravity which is called “General Relativity”. In brief, General Relativity is Einstein’s theory of gravity. In what follows, we will delve into General Relativity, showing how it is the curving of spacetime in the general theory that replaces the old Newtonian idea of gravity being just another force.
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Notes
- 1.
This paragraph can be skipped by non-calculus readers.
- 2.
These are frequently referred to as gedanken experiments from the German, meaning “thought” experiments, experiments not actually performed but rather imagined to have been performed. Einstein was particularly fond of such exercises. He employed them in his intellectual battles with N. Bohr over the issue of the probabilistic interpretation in quantum mechanics. See [5] for an interesting account of these exchanges.
- 3.
We also ignore the very tiny interactions between the rocks themselves.
- 4.
See for example [3] for an explicit demonstration of the changes to the form of the spacetime interval that are produced when the transformation to an accelerated reference frame is made.
- 5.
Note that for simplicity of expression, we have dropped the stars on the quantities that arise under the transformation to the new coordinate frame.
- 6.
The repetition of an index means that it is to be summed over \(0,1,2,3\). Here it is done for the indices \(i\) and \(k\). This is referred to as the “Einstein summation convention”. Note that the compact expression expands out to the form (4.4).
- 7.
Thus, \(x^i\) could represent \((r, \theta , \phi )\) spherical polar coordinates, \((r, z, \phi )\) cylindrical polar coordinates, etc.
- 8.
For mathematically inclined readers, the vector, which is the mathematical object of greater familiarity, is actually a special case of a tensor. Like soldiers in the army, tensors are categorized by their rank. Rank number is given by the number of indices attached to the tensor. A vector, having the single index \(i\), is a tensor of rank one. The metric tensor, having two indices, is a tensor of rank two. Tensors are mathematical constructs defined by their transformation properties. A nice development of the subject can be found in [7].
- 9.
A note for the calculus-equipped reader: These are actually partial derivatives of the metric tensor components.
- 10.
See Appendix B for a more detailed description of this equation.
- 11.
In Appendix B, we discuss the mathematics behind the intrinsic acceleration and how motion under gravity alone in General Relativity follows the geodesic equation.
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© 2012 Springer-Verlag Berlin Heidelberg
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Cooperstock, F.I., Tieu, S. (2012). Introducing Einstein’s General Relativity. In: Einstein's Relativity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30385-2_4
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DOI: https://doi.org/10.1007/978-3-642-30385-2_4
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